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                       SECURITIES AND EXCHANGE COMMISSION
                             WASHINGTON, D.C. 20549

                            ------------------------

                                   FORM 10-Q

[X] QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE SECURITIES EXCHANGE
    ACT OF 1934 FOR THE QUARTERLY PERIOD ENDED JUNE 30, 1998

                                       OR

[ ] TRANSITION REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE SECURITIES EXCHANGE
    ACT OF 1934 FOR THE TRANSITION PERIOD FROM _____________ TO _______________

                         COMMISSION FILE NUMBER: 0-6910

                            ------------------------

                               TEL OFFSHORE TRUST
             (Exact Name of Registrant as Specified in its Charter)

                 TEXAS                                           76-6004064
        (State of Incorporation,                              (I.R.S. Employer
            or Organization)                                 Identification No.)

          CHASE BANK OF TEXAS,
          NATIONAL ASSOCIATION
        CORPORATE TRUST DIVISION
            712 MAIN STREET
             HOUSTON, TEXAS                                         77002
         (Address of Principal                                   (Zip Code)
           Executive Offices)

       Registrant's Telephone Number, Including Area Code: (713) 216-5712

     Indicate by check mark whether the registrant (1) has filed all reports
required to be filed by Section 13 or 15(d) of the Securities Exchange Act of
1934 during the preceding 12 months (or for such shorter period that the
registrant was required to file such reports) and (2) has been subject to such
filing requirements for the past 90 days.  Yes [X]  No [ ]

     Indicate the number of shares outstanding of each of the registrant's
classes of common stock, as of the latest practicable date.

     As of August 6, 1998 -- 4,751,510 Units of Beneficial Interest in TEL
Offshore Trust.

================================================================================
<PAGE>
                   NOTE REGARDING FORWARD-LOOKING STATEMENTS

     This Form 10-Q includes "forward-looking statements" within the meaning
of Section 27A of the Securities Act of 1933, as amended, and Section 21E of the
Securities Exchange Act of 1934, as amended. All statements other than
statements of historical facts included in this Form 10-Q are forward-looking
statements. Although the Working Interest Owners have advised the Trust that
they believe that the expectations reflected in the forward-looking statements
contained herein are reasonable, no assurance can be given that such
expectations will prove to have been correct. Important factors that could cause
actual results to differ materially from expectations ("Cautionary
Statements") are disclosed in this Form 10-Q, including without limitation in
conjunction with the forward-looking statements included in this Form 10-Q. All
subsequent written and oral forward-looking statements attributable to the Trust
or persons acting on its behalf are expressly qualified in their entirety by the
Cautionary Statements.

                                       i

<PAGE>
                        PART I -- FINANCIAL INFORMATION

ITEM 1 -- FINANCIAL STATEMENTS

                               TEL OFFSHORE TRUST
               STATEMENTS OF ASSETS, LIABILITIES AND TRUST CORPUS

                                         JUNE 30,       DECEMBER 31,
                                           1998             1997
                                        ----------      ------------
                                        (UNAUDITED)
ASSETS
Cash and cash equivalents............   $2,381,691       $3,425,376
Net overriding royalty interest in
  producing oil and gas properties,
  net of accumulated amortization of
  $27,674,073 and
  $27,564,441, respectively..........      593,582          703,214
                                        ----------      ------------
Total assets.........................   $2,975,273       $4,128,590
                                        ==========      ============
LIABILITIES AND TRUST CORPUS
Distribution payable to Unit
  holders............................   $1,015,656       $1,952,687
Reserve for future Trust expenses....    1,366,035        1,472,689
Commitments and contingencies (Note
  7).................................
Trust corpus (4,751,510 Units of
  beneficial interest authorized and
  outstanding).......................      593,582          703,214
                                        ----------      ------------
Total liabilities and Trust corpus...   $2,975,273       $4,128,590
                                        ==========      ============

                       STATEMENTS OF DISTRIBUTABLE INCOME
                                  (UNAUDITED)
<TABLE>
<CAPTION>
                                           THREE MONTHS ENDED           SIX MONTHS ENDED
                                                JUNE 30,                    JUNE 30,
                                       --------------------------  --------------------------
                                           1998          1997          1998          1997
                                       ------------  ------------  ------------  ------------
<S>                                    <C>           <C>           <C>           <C>         
Royalty income.......................  $  1,115,995  $  2,051,820  $  3,050,387  $  3,177,825
Interest income......................        17,480        12,507        36,182        23,237
                                       ------------  ------------  ------------  ------------
                                          1,133,475     2,064,327     3,086,569     3,201,062
Decrease (increase) in reserve for
  future Trust expenses..............             0       (58,848)      106,654      (356,591)
General and administrative
  expenses...........................      (117,819)     (141,152)     (213,189)     (243,409)
                                       ------------  ------------  ------------  ------------
Distributable income.................  $  1,015,656  $  1,864,327  $  2,980,034  $  2,601,062
                                       ============  ============  ============  ============
Distributions per Unit (4,751,510
  Units).............................  $    .213754  $    .392365  $    .627175  $    .547417
                                       ============  ============  ============  ============

</TABLE>
   The accompanying notes are an integral part of these financial statements.

                                       1
<PAGE>
                               TEL OFFSHORE TRUST
                     STATEMENTS OF CHANGES IN TRUST CORPUS
                                  (UNAUDITED)
<TABLE>
<CAPTION>
                                             THREE MONTHS ENDED               SIX MONTHS ENDED
                                                  JUNE 30,                        JUNE 30,
                                       ------------------------------  ------------------------------
                                            1998            1997            1998            1997
                                       --------------  --------------  --------------  --------------
<S>                                    <C>             <C>             <C>             <C>           
Trust corpus, beginning of period....  $      639,536  $      911,004  $      703,214  $      938,589
Distributable income.................       1,015,656       1,864,327       2,980,034       2,601,062
Distribution payable to Unit
  holders............................      (1,015,656)     (1,864,327)     (2,980,034)     (2,601,062)
Amortization of net overriding
  royalty interest...................         (45,954)        (69,630)       (109,632)        (97,215)
                                       --------------  --------------  --------------  --------------
Trust corpus, end of period..........  $      593,582  $      841,374  $      593,582  $      841,374
                                       ==============  ==============  ==============  ==============

</TABLE>
   The accompanying notes are an integral part of these financial statements.

                                       2
<PAGE>
                               TEL OFFSHORE TRUST
                         NOTES TO FINANCIAL STATEMENTS
                                  (UNAUDITED)

NOTE 1 -- TRUST ORGANIZATION

     Tenneco Offshore Company, Inc. ("Tenneco Offshore") created the TEL
Offshore Trust ("Trust") effective January 1, 1983, pursuant to the Plan of
Dissolution ("Plan") approved by Tenneco Offshore's stockholders on December
22, 1982. In accordance with the Plan, the TEL Offshore Trust Partnership
("Partnership") was formed in which the Trust owns a 99.99% interest and
Tenneco Oil Company ("Tenneco") initially owned a .01% interest. In general,
the Plan was effected by transferring an overriding royalty interest
("Royalty") equivalent to a 25% net profits interest in the oil and gas
properties (the "Royalty Properties") of Tenneco Exploration, Ltd.
("Exploration I") located offshore Louisiana to the Partnership and issuing
certificates evidencing units of beneficial interest in the Trust ("Units") in
liquidation and cancellation of Tenneco Offshore's common stock.

     On October 31, 1986, Exploration I was dissolved and the oil and gas
properties of Exploration I were distributed to Tenneco subject to the Royalty.
Tenneco, who was then serving as the Managing General Partner of the
Partnership, assumed the obligations of Exploration I, including its obligations
under the instrument conveying the Royalty to the Partnership (the
"Conveyance"). The dissolution of Exploration I had no impact on future cash
distributions to Unit holders.

     On November 18, 1988, Chevron U.S.A. Inc. ("Chevron") acquired most of
the Gulf of Mexico offshore oil and gas properties of Tenneco, including all the
Royalty Properties. As a result of the acquisition, Chevron replaced Tenneco as
the Working Interest Owner and Managing General Partner of the Partnership.
Chevron also assumed Tenneco's obligations under the Conveyance.

     On October 30, 1992, Pennzoil Company ("Pennzoil") acquired certain oil
and gas producing properties from Chevron, including four of the Royalty
Properties. The four Royalty Properties acquired by Pennzoil were East Cameron
354, Eugene Island 348, Eugene Island 367 and Eugene Island 208. As a result of
such acquisition, Pennzoil replaced Chevron as the Working Interest Owner of
such properties on October 30, 1992. Pennzoil also assumed Chevron's obligations
under the Conveyance with respect to such properties.

     On December 1, 1994, Texaco Exploration and Production Inc. ("Texaco")
acquired one of the Royalty Properties from Chevron. The Royalty Property
acquired by Texaco was West Cameron 643. As a result of such acquisition, Texaco
replaced Chevron as the Working Interest Owner of such property on December 1,
1994. Texaco also assumed Chevron's obligations under the Conveyance with
respect to such property.

     On October 1, 1995, SONAT Exploration Company ("SONAT") acquired the East
Cameron 354 property from Pennzoil. In addition, on October 1, 1995, Amoco
Production Company ("Amoco") acquired the Eugene Island 367 property from
Pennzoil. As a result of such acquisitions, SONAT and Amoco have replaced
Pennzoil as the Working Interest Owners of the East Cameron 354 and Eugene
Island 367 properties, respectively, on October 1, 1995 and also assumed
Pennzoil's obligations under the Conveyance with respect to such properties.

     Effective January 1, 1998, Energy Resource Technology, Inc. ("Energy")
acquired the East Cameron 354 property from SONAT. As a result of such
acquisition, Energy replaced SONAT as the Working Interest Owner of the East
Cameron 354 property effective January 1, 1998, and also assumed SONAT's
obligations under the Conveyance with respect to such property.

     All of the Royalty Properties continue to be subject to the Royalty, and it
is anticipated that the Trust and the Partnership, in general, will continue to
operate as if the above-described sales of the Royalty Properties had not
occurred.

                                       3
<PAGE>
                               TEL OFFSHORE TRUST
                  NOTES TO FINANCIAL STATEMENTS -- (CONTINUED)
                                  (UNAUDITED)

     Unless the context in which such terms are used indicates otherwise, in
these Notes the terms "Working Interest Owner" and "Working Interest Owners"
generally refer to the owner or owners of the Royalty Properties (Exploration I
through October 31, 1986; Tenneco for periods from October 31, 1986 until
November 18, 1988; Chevron with respect to all Royalty Properties for periods
from November 18, 1988 until October 30, 1992, and with respect to all Royalty
Properties except East Cameron 354, Eugene Island 348, Eugene Island 367 and
Eugene Island 208 for periods from October 30, 1992 until December 1, 1994, and
with respect to the same properties except West Cameron 643 thereafter; Pennzoil
with respect to East Cameron 354, Eugene Island 348, Eugene Island 367 and
Eugene Island 208 for periods from October 30, 1992 until October 1, 1995, and
with respect to Eugene Island 348 and Eugene Island 208 thereafter; Texaco with
respect to West Cameron 643 for periods beginning on or after December 1, 1994;
SONAT with respect to East Cameron 354 for periods beginning on or after October
1, 1995; and Amoco with respect to Eugene Island 367 for periods beginning on or
after October 1, 1995; and Energy with respect to East Cameron 354 for periods
beginning on or after January 1, 1998).

NOTE 2 -- BASIS OF ACCOUNTING

     The accompanying unaudited financial information has been prepared by Chase
Bank of Texas, National Association ("Corporate Trustee") in accordance with
the instructions to Form 10-Q and does not include all of the information
required by generally accepted accounting principles for complete financial
statements, although the Corporate Trustee and the individual trustees
(collectively, the "Trustees") believe that the disclosures are adequate to
make the information presented not misleading. The information furnished
reflects all adjustments which are, in the opinion of the Trustees, necessary
for a fair presentation of the results for the interim periods presented. The
financial information should be read in conjunction with the financial
statements and notes thereto included in the Trust's Annual Report on Form 10-K
for the year ended December 31, 1997.

     The financial statements of the Trust are prepared on the following basis:

          (a)  Royalty income is recorded when received by the Corporate Trustee
               on the last business day of each calendar quarter; and

          (b)  Trust general and administrative expenses are recorded when paid,
               except for the cash reserved for future general and
               administrative expenses, as discussed in Note 6.

     This manner of reporting income and expenses is considered to be the most
meaningful because the quarterly distributions to Unit holders are based on net
cash receipts received from the Working Interest Owners. The financial
statements of the Trust differ from financial statements prepared in accordance
with generally accepted accounting principles, because, under such principles,
Royalty income and Trust general and administrative expenses for a quarter would
be recognized on an accrual basis. In addition, amortization of the overriding
royalty interest, which is calculated based on units-of-production, is charged
directly to Trust corpus since such amount does not affect distributable income.

     Cash and cash equivalents include all highly liquid short term investments
with original maturities of three months or less.

NOTE 3 -- OVERRIDING ROYALTY INTEREST

     The Royalty entitles the Trust to its share (99.99%) of 25% of the Net
Proceeds attributable to the Royalty Properties. The Conveyance, dated January
1, 1983, provides that the Working Interest Owners will calculate, for each
period of three months commencing the first day of February, May, August and
November, an amount equal to 25% of the Net Proceeds from their oil and gas
properties for the period.

                                       4
<PAGE>
                               TEL OFFSHORE TRUST
                  NOTES TO FINANCIAL STATEMENTS -- (CONTINUED)
                                  (UNAUDITED)

Generally, Net Proceeds are the amounts received by the Working Interest Owners
from the sale of minerals from the Royalty Properties less operating and capital
costs incurred, management fees and expense reimbursements owing the Managing
General Partner of the Partnership, applicable taxes other than income taxes,
and cash escrows. Cash escrows are for the future costs to be incurred to plug
and abandon wells, dismantle and remove platforms, pipelines and other
production facilities, and for the estimated amount of future capital
expenditures on the Royalty Properties. Net Proceeds do not include amounts
received by the Working Interest Owners as advance gas payments, "take-or-pay"
payments or similar payments unless and until such payments are extinguished or
repaid through the future delivery of gas.

NOTE 4 -- DISTRIBUTIONS TO UNIT HOLDERS

     In accordance with the provisions of the Trust Agreement, generally all Net
Proceeds received by the Trust, net of Trust general and administrative expenses
and any cash reserves established for the payment of contingent or future
obligations of the Trust, are distributed currently to the Unit holders. The
amounts distributed are determined on a quarterly basis and are payable to Unit
holders of record as of the last business day of each calendar quarter. However,
cash distributions are made in January, April, July and October and include
interest earned from the quarterly record date to the date of distribution.

NOTE 5 -- SPECIAL COST ESCROW ACCOUNT

     The Conveyance provides for reserving funds for estimated future "Special
Costs" of plugging and abandoning wells, dismantling platforms and other costs
of abandoning the Royalty Properties, as well as for the estimated amount of
future drilling projects and other capital expenditures on the Royal Properties.
As provided in the Conveyance, the amount of funds to be reserved is determined
based on factors including estimates of aggregate future production costs,
aggregate future Special Costs, aggregate future net revenues and actual current
net proceeds. Deposits into this account reduce current distributions and are
placed in an escrow account and invested in short-term certificates of deposit.
Such account is herein referred to as the "Special Cost Escrow Account." The
Trust's share of interest generated from the Special Cost Escrow Account serves
to reduce the Trust's share of allocated production costs. Special Cost Escrow
funds will generally be utilized to pay Special Costs to the extent there are
not adequate current net proceeds to pay such costs. Special Costs that have
been paid are no longer included in the Special Cost Escrow calculation.
Deposits to the Special Cost Escrow Account will generally be made when the
balance in the Special Cost Escrow Account is less than 125% of future Special
Costs and there is a Net Revenues Shortfall (a calculation of the excess of
estimated future costs over estimated future net revenues pursuant to a formula
contained in the Conveyance). When there is not a Net Revenues Shortfall,
amounts in the Special Cost Escrow Account will generally be released, to the
extent that Special Costs have been incurred. Amounts in the Special Cost Escrow
Account generally will also be released when the balance in such account exceeds
125% of future Special Costs. In the first six months of 1997, there was a net
deposit of funds into the Special Cost Escrow Account. The Trust's share of the
funds deposited was approximately $778,000. The deposit was primarily a result
of an increase in the current estimate of projected capital expenditures,
production costs and abandonment costs in connection with the West Cameron 643
drilling in 1996. In the first six months of 1998, there was a net release of
funds from the Special Cost Escrow Account. The Trust's share of the funds
released was approximately $1,322,700. The release was primarily a result of a
decrease in the current estimate of projected capital expenditures of the
Royalty Properties. As of June 30, 1998, approximately $3,299,600 (net to the
Trust) remained in the Special Cost Escrow Account.

                                       5
<PAGE>
                               TEL OFFSHORE TRUST
                  NOTES TO FINANCIAL STATEMENTS -- (CONTINUED)
                                  (UNAUDITED)

     Chevron, in its capacity as Managing General Partner, has advised the Trust
that additional deposits to the Special Cost Escrow Account may be required in
future periods in connection with other production costs, other abandonment
costs, other capital expenditures and changes in the estimates and factors
described above. Such deposits could result in a significant reduction in
Royalty income in the periods in which such deposits are made, including the
possibility that no Royalty income would be received in such periods.

NOTE 6 -- EXPENSE RESERVE

     At December 31, 1991, a cash reserve of $120,000 had been established for
future Trust general and administrative expenses. During 1992 and 1993, in
anticipation of future periods when the cash received from the Royalty may not
be sufficient for payment of Trust expenses, the reserve for future Trust
general and administrative expenses was increased each quarter by an amount
equal to the difference between $150,000 and the amount of the Trust's general
and administrative expenses for such quarter. In March 1994, the Trust
determined, in accordance with the Trust Agreement, to begin further increasing
the Trust's cash reserve each quarter by an amount equal to the difference
between $200,000 and the amount of the Trust's general and administrative
expenses for such quarter. During 1996, the Trust used net cash of $298,309 from
the Trust's cash reserve account to pay the Trust's general and administrative
expenses, due to the absence of Royalty income in the first, second and fourth
quarters. During 1997, the aggregate amount of cash reserved by the Trust was
$593,066. In the first quarter of 1998, the Trust determined that the Trust's
cash reserve was currently sufficient to provide for future administrative
expenses in connection with the winding up of the Trust. The Trust determined
that a cash reserve equal to three times the average expenses of the Trust
during each of the past three fiscal years was sufficient at this time to
provide for future administrative expenses in connection with the winding up of
the Trust. This reserve amount for 1998 is $1,366,035. The excess amount of
$106,654 was distributed to Unit holders in the first quarter of 1998, and no
deposit was made in the second quarter of 1998. No deposits are expected to be
made to the Trust's cash reserve account during 1998.

NOTE 7 -- COMMITMENTS AND CONTINGENCIES

     During 1994, Pennzoil, the Working Interest Owner on the Eugene Island 348
property, settled a gas imbalance on that property for approximately $2,696,000.
The Trust's share of this settlement amount was approximately $674,000, of which
approximately $361,000 has been recovered from the Trust by the Working Interest
Owner through the second quarter of 1998, and the remainder will be subject to
recovery from the Trust during future periods in accordance with the Conveyance.
The Working Interest Owner has advised the Trust that future Royalty income
attributable to all of the Royalty Properties owned by Pennzoil will be used to
offset the Trust's share of such settlement amounts. Based on current
production, prices and expenses for the Royalty Properties owned by Pennzoil, it
is estimated that Royalty income attributable to such properties will be
retained by Pennzoil for the remaining life of the Trust. The Trust does not
anticipate that retention of such Royalty income by Pennzoil will have a
material effect on the Trust's Royalty income as a whole.

     The Working Interest Owners have advised the Trust that, although they
believe that they are in general compliance with applicable health, safety and
environmental laws and regulations that have taken effect at the federal, state
and local levels, costs may be incurred to comply with current and proposed
environmental legislation which could result in increased operating expenses on
the Royalty Properties.

                                       6

<PAGE>
ITEM 2.  MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION AND
RESULTS OF OPERATIONS

FINANCIAL REVIEW

THREE MONTHS ENDED JUNE 30, 1998 AND 1997

     Distributions to Unit holders for the three months ended June 30, 1998
amounted to $1,015,656 or $.213754 per Unit as compared to $1,864,327 or
$.392365 per Unit for the same period in 1997. The decrease in distributable
income for the second quarter of 1998 was primarily due to a significant
decrease in crude oil and condensate revenues in the second quarter of 1998, as
compared to the second quarter of 1997.

     Crude oil and condensate revenues decreased approximately 46% in the second
quarter of 1998 in comparison to the same period in 1997 primarily due to a 29%
decrease in the average price received from $19.56 per barrel in the second
quarter of 1997 to $13.97 per barrel in the second quarter of 1998. In addition,
there was a 24% decrease in crude oil and condensate volumes from the 1997
second quarter to the 1998 second quarter. This decrease was primarily
attributable to decreased production from the B-11, B-12 and B-13 wells on the
Ship Shoal 182/183 property in the second quarter of 1998. Gas revenues
decreased approximately 12% in the second quarter of 1998 compared to the second
quarter of 1997 primarily due to a 14% decrease in gas volumes. This decrease
was primarily attributable to higher production from the B-9 well on the West
Cameron 643 property in the second quarter of 1997. This decrease in volumes was
offset by a 3% increase in the average price received for natural gas from $2.23
per Mcf in the second quarter of 1997 to $2.30 per Mcf in the second quarter of
1998. The Trust's share of capital expenditures increased by approximately 3380%
or $1,265,328 in the second quarter of 1998 as compared to the same period in
1997 primarily due to costs incurred in the second quarter of 1998 which were
associated with drilling the B-7, B-9 and B-12 wells in the first quarter of
1998 and the B-16 well in the second quarter of 1998 on the Eugene Island 339
property. The Trust's share of operating expenses decreased by approximately 24%
or $92,708 in the second quarter of 1998 as compared to the same period in 1997
primarily due to service facility charges in the second quarter of 1997.

     For the second quarter of 1998, the Trust had undistributed net income of
$1,645. Undistributed net income represents positive Net Proceeds generated
during the period that were applied to an existing loss carryforward. The
undistributed net income for the second quarter of 1998 was applied to a loss
carryforward that resulted primarily from the Eugene Island 348 gas imbalance
settlement in 1994. See Note 7 in the Notes to Financial Statements for
information regarding such settlement.

     In the second quarter of 1998, there was a net release of funds from the
Special Cost Escrow Account. The Trust's share of the funds released was
approximately $808,900, compared to a deposit of funds into the Special Cost
Escrow Account of approximately $434,000 net to the Trust in the second quarter
of 1997. The Special Cost Escrow is set aside for estimated abandonment costs
and future capital expenditures as provided for in the Conveyance. For
additional information relating to the Special Cost Escrow see "Special Cost
Escrow Account" below.

SIX MONTHS ENDED JUNE 30, 1998 AND 1997

     Distributions to Unit holders for the six months ended June 30, 1998
amounted to $2,980,034 or $.627175 per Unit as compared to $2,601,062 or
$.547417 per Unit for the same period in 1997. The increase in distributable
income for the first six months of 1998 was primarily due to a net release of
approximately $1,322,700 from the Trust's Special Cost Escrow Account as
compared to a net deposit of approximately $778,000 in the first six months of
1997.

                                       7
<PAGE>
     Crude oil and condensate revenues decreased approximately 34% in the first
six months of 1998 in comparison to the same period in 1997 primarily due to a
26% decrease in the average price received from $21.19 per barrel for the six
months ended June 30, 1997 to $15.66 per barrel for the six months ended June
30, 1998. In addition, there was a 10% decrease in crude oil and condensate
volumes. This decrease was primarily attributable to decreased production from
the B-11, B-12 and B-13 wells on the Ship Shoal 182/183 property in the first
six months of 1998. Gas revenues decreased 34% in the first six months of 1998
compared to the first six months of 1997 primarily due to a 23% decrease in gas
volumes, which decrease was primarily attributable to higher production from the
B-9 well on the West Cameron 643 property in the second quarter of 1997 and the
B-15 well watering out on the Ship Shoal 182/183 property in the first quarter
of 1998. In addition, there was a 13% decrease in the average price received for
natural gas from $2.91 per Mcf in the first six months of 1997 to $2.52 per Mcf
in the first six months of 1998. The Trust's share of capital expenditures
increased by approximately 194% or $1,382,119 for the six months ended June 30,
1998 as compared to the same period in 1997 primarily due to the costs
associated with drilling the B-7, B-9 and B-12 wells in the first quarter of
1998 and the B-16 well in the second quarter of 1998 on the Eugene Island 339
property. The Trust's share of operating expenses decreased by approximately 53%
or $449,516 for the six months ended June 30, 1998 as compared to the same
period in 1997 primarily due to a workover on the E-9 well on the Ship Shoal
182/183 property in the first quarter of 1997.

     For the first six months of 1998, the Trust had undistributed net income of
$34,110. Undistributed net income represents positive Net Proceeds generated
during the period that were applied to an existing loss carryforward. The
undistributed net income for the the first six months of 1998 was applied to a
loss carryforward that resulted primarily from the Eugene Island 348 gas
imbalance settlement in 1994. See Note 7 in the Notes to Financial Statements
for information regarding such settlement.

     In the first six months of 1998, there was a net release of funds from the
Special Cost Escrow Account. The Trust's share of the funds released was
approximately $1,322,700, compared to a net deposit of funds into the Special
Cost Escrow Account of $778,000 net to the Trust in the first six months of
1997. For additional information relating to the Special Cost Escrow see
"Special Cost Escrow Account" below.

RESERVE FOR FUTURE TRUST EXPENSES

     In accordance with the provisions of the Trust Agreement, generally all Net
Proceeds received by the Trust, net of Trust general and administrative expenses
and any cash reserves established for the payment of contingent or future
obligations of the Trust, are distributed currently to the Unit holders. During
1992 and 1993, in anticipation of future periods when the cash received from the
Royalty may not be sufficient for payment of Trust expenses, the reserve for
future Trust general and administrative expenses was increased each quarter by
an amount equal to the difference between $150,000 and the amount of the Trust's
general and administrative expenses for such quarter. In March 1994, the Trust
determined, in accordance with the Trust Agreement, to begin further increasing
the Trust's cash reserve each quarter by an amount equal to the difference
between $200,000 and the amount of the Trust's general and administrative
expenses for such quarter. During 1996, the Trust used net cash of $298,309 from
the Trust's cash reserve account to pay the Trust's general and administrative
expenses, due to the absence of Royalty income during the first, second and
fourth quarters. During 1997, the aggregate amount of cash reserved by the Trust
was $593,066. The total amount of the Trust's cash reserve at December 31, 1997
was $1,472,689. In addition, in the first quarter of 1998, the Trust determined
that the Trust's cash reserve was currently sufficient to provide for future
administrative expenses in connection with the winding up of the Trust. The
Trust determined that a cash reserve equal to three times the average expenses
of the Trust during each of the past three fiscal years was sufficient at this
time to provide for future administrative expenses in connection with the
winding up of the Trust. This reserve amount for 1998 is $1,366,035. The excess
amount of $106,654 was distributed to

                                       8
<PAGE>
Unit holders in the first quarter of 1998, and no deposit was made in the second
quarter of 1998. No deposits are expected to be made to the Trust's cash reserve
account during 1998.

OTHER

     The amount of cash distributed by the Trust is dependent on, among other
things, the sales prices and quantities of oil and gas produced from the Royalty
Properties. It should be noted that substantial uncertainties exist with regard
to future oil and gas prices, which are subject to material fluctuations due to
changes in production levels and pricing and other actions taken by major
petroleum producing nations, as well as the regional supply and demand for oil
and gas, weather, industrial growth, conservation measures, competition and
other variables.

OPERATIONAL REVIEW

THREE MONTHS ENDED JUNE 30, 1998 AND 1997

     VOLUMES AND DOLLAR AMOUNTS DISCUSSED BELOW REPRESENT AMOUNTS RECORDED BY
THE WORKING INTEREST OWNERS UNLESS OTHERWISE SPECIFIED.

     Ship Shoal 182/183 crude oil revenues decreased from $6,896,637 in the
second quarter of 1997 to $3,492,285 in the second quarter of 1998, primarily
due to a decrease in the average crude oil price from $20.14 per barrel in the
second quarter of 1997 to $14.61 per barrel for the same period in 1998. In
addition, there was a decrease in crude oil production from 342,423 barrels in
the second quarter of 1997 to 239,033 barrels in the second quarter of 1998. The
decrease in crude oil production was due primarily to lower production in 1998
on the B-11, B-12 and B-13 wells that were drilled in 1996. Gas revenues
decreased from $908,641 in the second quarter of 1997 to $907,200 in the second
quarter of 1998 primarily due to a decrease in gas volumes from 434,507 Mcf in
the second quarter of 1997 to 401,622 Mcf in the second quarter of 1998. The
decrease in gas volumes was also primarily due to lower production in 1998 on
the B-11, B-12 and B-13 wells. The decrease in gas volumes was partially offset
by an increase in the average natural gas sales price from $2.09 per Mcf in the
second quarter of 1997 to $2.34 per Mcf in the same period of 1998. The majority
of the gas from this property is currently being purchased by Dynegy Inc.
("Dynegy") at a calculated price based on the monthly FERC Tennessee-Louisiana
Zone 1 index. In addition, the Working Interest Owner has advised the Trust that
approximately 59,405 Mcf have been overtaken by the Working Interest Owner from
this property as of April 30, 1998. The Trust's share of this overtake position
is approximately 14,851 Mcf. Accordingly, gas revenues from this property may be
reduced in future periods while underproduced parties recover their share of the
gas imbalance. Capital expenditures increased $104,978 for the second quarter of
1998 as compared to the same period in 1997. Operating expenses decreased from
$493,348 in the second quarter of 1997 to $306,284 for the same period of 1998
due primarily to a reduction in drilling activity in 1998. The Working Interest
Owner has advised the Trust that it drilled the E-10 well on this property in
April 1998 at an approximate cost of $3.7 million ($925,000 net to the Trust).
This well is currently producing approximately 334 Mcf per day.

     Eugene Island 339 crude oil revenues decreased from $1,304,829 in the
second quarter of 1997 to $942,231 in the second quarter of 1998 due primarily
to a decrease in the average crude oil price from $16.95 per barrel in the
second quarter of 1997 to $12.02 per barrel in the second quarter of 1998. This
decrease in the average crude oil price was slightly offset by an increase in
volumes from 76,970 barrels in the second quarter of 1997 to 78,375 barrels in
the second quarter of 1998. Gas revenues increased from $137,527 in the second
quarter of 1997 to $331,606 in the second quarter of 1998 due to an increase in
gas volumes from 62,668 Mcf in the second quarter of 1997 to 149,243 Mcf for the
same period in 1998. The increase in gas volumes was due primarily to a well
being shut down during the second quarter of 1997 for compressor repair. In
addition, there was an increase in the average price received for natural gas
from

                                       9
<PAGE>
$2.21 per Mcf in the second quarter of 1997 to $2.40 per Mcf in the second
quarter of 1998. The Working Interest Owner has advised the Trust that there is
an undertake imbalance position of approximately 68,812 Mcf on this property as
of April 30, 1998. The Trust's share of this undertake position is approximately
17,203 Mcf. Accordingly, gas revenues from this property may be increased in
future periods while overproduced parties release their share of the gas
imbalance. Chevron has advised the Trust that sufficient gas reserves exist on
the Eugene Island 339 for underproduced parties to recoup their share of the gas
imbalance on this property. The gas from this property is currently committed to
Dynegy pursuant to an agreement providing for gas to be purchased at a
calculated price based on the monthly Inside FERC Tennessee-Louisiana Zone 1
index. Capital expenditures increased $3,932,476 from second quarter 1997 to
second quarter 1998 due primarily to costs incurred in the second quarter of
1998 which were associated with drilling activity on the B-7, B-9 and B-12 wells
in the first quarter of 1998 and the B-16 well in the second quarter of 1998.
Operating expenses decreased from $590,672 in the second quarter of 1997 to
$127,479 for the same period in 1998 primarily due to service facility charges
in the second quarter of 1997. The Working Interest Owner has advised the Trust
that the drilling results of the B-7, B-9 and B-12 wells in the first quarter of
1998 were disappointing. Also, the Working Interest Owner has advised the Trust
that it completed the B-4 sidetrack well in March 1998 at an aggregate cost of
approximately $480,000 ($120,000 net to the Trust).

     West Cameron 643 gas revenues decreased from $2,351,044 in the second
quarter of 1997 to $1,701,139 in the second quarter of 1998 due primarily to a
decrease in gas volume from 1,053,600 Mcf in the second quarter of 1997 to
744,366 Mcf for the same period in 1998. The decrease in gas volumes was due
primarily to the successful B-8 and B-9 wells drilled on this property in the
second quarter of 1996 and the successful workovers on the A-2 and A-9 wells in
the first quarter of 1996. The decrease in volumes was slightly offset by an
increase in the average price received for natural gas from $2.23 per Mcf in the
second quarter of 1997 to $2.29 per Mcf for the same period of 1998. The Working
Interest Owner has advised the Trust that the gas from this property is
currently committed under the contract with Texaco Natural Gas, Inc. pursuant to
an agreement for gas to be purchased at a price based on the monthly Inside FERC
Tennessee-Louisiana Zone 1 index. Capital expenditures increased $378,487 for
the second quarter of 1998 as compared to the same period in 1997 due primarily
to a credit for returned equipment in the second quarter of 1997 and increased
rig activity due to the sidetrack of the A-10 and A-14 wells in the second
quarter of 1998. Operating expenses increased from $339,791 in the second
quarter of 1997 to $613,922 for the same period in 1998 due primarily to a
workover on the A-6 well and repairs on an air conditioner unit in the second
quarter of 1998.

     In June 1998, Chevron advised the Trust that production began in May 1998
on the two wells on the East Cameron 371 property. The Trust has been advised by
the Working Interest Owner on this property that the well is currently producing
approximately 75,000 Mcf of gas per day and approximately 49,000 barrels of
crude oil and condensate per day.

SIX MONTHS ENDED JUNE 30, 1998 AND 1997

     Volumes and dollar amounts discussed below represent amounts recorded by
the Working Interest Owners unless otherwise specified.

     Ship Shoal 182/183 crude oil revenues decreased from $12,569,251 in the
first six months of 1997 to $8,120,679 in the first six months of 1998,
primarily due to a decrease in the average crude oil price from $21.64 per
barrel in the first six months of 1997 to $16.21 per barrel for the same period
in 1998. In addition, there was a decrease in crude oil production from 580,713
barrels in the first six months of 1997 to 500,951 barrels in the first six
months of 1998. The decrease in crude oil production was due primarily to the
lower production in 1998 on the B-11, B-12 and B-13 wells that were drilled in
1996. Gas revenues decreased from $2,259,299 in the first six months of 1997 to
$1,790,444 in the first six months of 1998

                                       10
<PAGE>
primarily due to a decrease in the average natural gas sales price from $2.82
per Mcf in the first six months of 1997 to $2.56 per Mcf in the same period of
1998. In addition, there was a decrease in gas volumes from 812,010 Mcf in the
first six months of 1997 to 731,812 Mcf in the first six months of 1998. The
decrease in gas volumes was primarily due to lower production in 1998 on the
B-11, B-12 and B-13 wells and the B-15 well watering out in the first quarter of
1998. Capital expenditures decreased from $2,170,816 in the first six months of
1997 to $220,006 in the first six months of 1998 due primarily to capital
expenditures recognized in 1997 from the drilling and completion of the B-11,
B-12 and B-13 wells in 1996. Operating expenses decreased from $1,235,952 for
the six months of 1997 to $731,592 for the same period in 1998 due primarily to
a reduction in drilling activity in 1998 and the workover on the E-9 well in the
first quarter of 1997.

     Eugene Island 339 crude oil revenues decreased from $3,028,695 in the first
six months of 1997 to $2,104,080 in the first six months of 1998 due primarily
to a decrease in the average crude oil price from $19.51 per barrel in the first
six months of 1997 to $13.80 per barrel in the first six months of 1998. In
addition, there was a slight decrease in volumes from 155,227 barrels in the
first six months of 1997 to 152,478 barrels for the same period in 1998. Gas
revenues increased from $513,948 in the first six months of 1997 to $628,450
primarily due to an increase in gas volumes from 175,368 Mcf in the first six
months of 1997 to 259,945 Mcf for the same period in 1998. The increase in gas
volumes was due primarily to a well being shut down during the first and second
quarter of 1997 for upgrading the facility and compressor repair. The increase
in gas volumes was offset by a decrease in the average price received for
natural gas from $2.98 per Mcf in the first six months of 1997 to $2.62 per Mcf
in the first six months of 1998. Operating expenses decreased from $1,031,519
for the first six months of 1997 to $265,886 for the first six months of 1998
due primarily to service facility charges in the second quarter of 1997. Capital
expenditures increased $5,386,884 in comparison from the first six months of
1997 to the first six months of 1998 due primarily to drilling activity on the
B-7, B-9 and B-12 wells in the first quarter of 1998.

     West Cameron 643 gas revenues decreased from $6,931,646 in the first six
months of 1997 to $3,691,224 in the first six months of 1998 due primarily to a
decrease in gas volumes from 2,364,726 Mcf in the first six months of 1997 to
1,482,543 Mcf for the same period in 1998. The decrease in gas volumes was due
primarily to the successful B-8 and B-9 wells drilled on this property in the
second quarter of 1996 and the successful workovers in the A-2 and A-9 wells in
the first quarter of 1996. In addition, there was a decrease in the average
price received for natural gas from $2.93 per Mcf in the first six months of
1997 to $2.49 per Mcf for the same period of 1998. Capital expenditures
increased from $367,659 in the first six months of 1997 to $504,841 in the first
six months of 1998 due primarily to a credit for returned equipment in the
second quarter of 1997 and increased rig activity due to the sidetrack of the
A-10 and A-14 wells in the second quarter of 1998. Operating expenses decreased
from $880,220 in the first six months of 1997 to $492,871 for the same period in
1998 due primarily to lower costs as a result of the removal of a compressor
from the property in the first quarter of 1998.

FUTURE NET REVENUES AND TERMINATION OF THE TRUST

     Based on a reserve study provided to the Trust by DeGolyer and MacNaughton,
independent petroleum engineers, it was estimated that as of October 31, 1997
future net revenues attributable to the Trust's royalty interests approximated
$24.3 million (subject to correction as described below). Such reserve study
also indicates that approximately 80% of the future net revenues from the
Royalty Properties are expected to be received by the Trust during the next 3
years. In addition, because the Trust will terminate in the event estimated
future net revenues fall below $2 million, it would be possible for the Trust to
terminate even though some or all of the Royalty Properties continued to have
remaining productive lives. Upon termination of the Trust, the Trustees will
sell for cash all of the assets held in the Trust estate and make a

                                       11
<PAGE>
final distribution to Unit holders of any funds remaining after all Trust
liabilities have been satisfied. The estimates of future net revenues discussed
above are subject to large variances from year to year and should not be
construed as exact. There are numerous uncertainties present in estimating
future net revenues for the Royalty Properties. The estimate may vary depending
on changes in market prices for crude oil and natural gas, the recoverable
reserves, annual production and costs assumed by DeGolyer and MacNaughton. In
addition, future economic and operating conditions as well as results of future
drilling plans may cause significant changes in such estimate. The discussion
set forth above is qualified in its entirety by reference to the Trust's 1997
Annual Report on Form 10-K. The Form 10-K is available upon request from the
Corporate Trustee.

     The Trust has been recently advised by the Working Interest Owner that the
interest with respect to the East Cameron 371 property provided to DeGolyer & 
MacNaughton was incorrect and therefore, the calculation included in the
reserve report as of October 31, 1997 should be revised. The calculation used
by DeGolyer & MacNaughton was based on a combined interest in this property of
1.875%, rather than net to the Trust of 1.875%. Accordingly, the Trust's
interest in this property appears to have been understated by 75%. DeGolyer &
MacNaughton has not yet completed the revision of its reserve report, but the
Trust has been informed that DeGolyer & MacNaughton anticipates it will provide
the Trust with a copy of a corrected reserve report prior to August 28, 1998.
The Trust will file an amendment to this Form 10-Q, the Trust's Form 10-K for
the year ended December 31, 1997 and the Trust's Form 10-Q for the three months
ended March 31, 1998 to reflect any changes caused by the correction of the
reserve report by DeGolyer & MacNaughton. DeGolyer & MacNaughton currently
anticipates such calculation will not cause an increase in future net revenues
net to the Trust as of October 31, 1997 in excess of $2 million, or 8% of the
$24.3 million previously reported in their report.


SPECIAL COST ESCROW ACCOUNT

     The Conveyance provides for reserving funds for estimated future "Special
Costs" of plugging and abandoning wells, dismantling platforms and other costs
of abandoning the Royalty Properties, as well as for the estimated amount of
future drilling projects and other capital expenditures on the Royalty
Properties. As provided in the Conveyance, the amount of funds to be reserved is
determined based on factors including estimates of aggregate future production
costs, aggregate future Special Costs, aggregate future net revenues and actual
current net proceeds. Deposits into this account reduce current distributions
and are placed in an escrow account and invested in short-term certificates of
deposit. Such account is herein referred to as the "Special Cost Escrow
Account." The Trust's share of interest generated from the Special Cost Escrow
Account serves to reduce the Trust's share of allocated production costs.
Special Cost Escrow funds will generally be utilized to pay Special Costs to the
extent there are not adequate current net proceeds to pay such costs. Special
Costs that have been paid are no longer included in the Special Cost Escrow
calculation. Deposits to the Special Cost Escrow Account will generally be made
when the balance in the Special Cost Escrow Account is less than 125% of future
Special Costs and there is a Net Revenues Shortfall (a calculation of the excess
of estimated future costs over estimated future net revenues pursuant to a
formula contained in the Conveyance). When there is not a Net Revenues
Shortfall, amounts in the Special Cost Escrow Account will generally be
released, to the extent that Special Costs have been incurred. Amounts in the
Special Cost Escrow Account generally will also be released when the balance in
such account exceeds 125% of future Special Costs. In the first six months of
1997, there was a net deposit of funds into the Special Cost Escrow Account. The
Trust's share of the funds deposited was approximately $778,000. The deposit was
primarily a result of an increase in the current estimates of projected capital
expenditures, production costs and abandonment costs in connection with the West
Cameron 643 drilling in 1996. In the first six months of 1998, there was a net
release of funds from the Special Cost Escrow Account. The Trust's share of the
funds released was approximately $1,322,700. The release was primarily a result
of a decrease in the current estimate of projected capital expenditures of the
Royalty Properties. As of June 30, 1998, approximately $3,299,600 (net to the
Trust) remained in the Special Cost Escrow Account.

     Chevron, in its capacity as Managing General Partner, has advised the Trust
that additional deposits to the Special Cost Escrow Account may be required in
future periods in connection with other production costs, other abandonment
costs, other capital expenditures and changes in the estimates and factors
described above. Such deposits, if made, could result in a significant reduction
in Royalty income in the periods in which such deposits are made, including the
possibility that no Royalty income would be received in such periods.

                                       12
<PAGE>
OVERVIEW OF PRODUCTION, PRICES AND NET PROCEEDS

     The following schedule provides a summary of the volumes and weighted
average prices for crude oil and condensate and natural gas recorded by the
Working Interest Owners for the Royalty Properties, as well as the Working
Interest Owners' calculations of the net proceeds and the royalties paid to the
Trust during the periods indicated. Net proceeds due to the Trust are calculated
for each three month period commencing on the first day of February, May, August
and November.

                                                ROYALTY PROPERTIES
                                                THREE MONTHS ENDED
                                                   JUNE 30,(1)
                                          ------------------------------
                                               1998            1997
                                          --------------  --------------
Crude oil and condensate (bbls).........         320,510         422,338
Natural gas and gas products (Mcf)......       1,470,067       1,699,755
Crude oil and condensate average price,
  per bbl...............................  $        13.97  $        19.56
Natural gas average price, per Mcf
  (excluding gas products)..............  $         2.30  $         2.23
Crude oil and condensate revenues.......  $    4,478,883  $    8,259,693
Natural gas and gas products revenues...       3,322,560       3,784,703
Production expenses.....................      (1,354,793)     (1,757,162)
Capital expenditures....................      (5,211,037)       (149,727)
Undistributed Net Loss (Income)(2)......          (6,579)       (193,615)
(Provision for) Refund of escrowed
  special costs.........................       3,235,394      (1,735,792)
                                          --------------  --------------
NET PROCEEDS............................       4,464,428       8,208,100
Royalty interest........................            x25%            x25%
                                          --------------  --------------
Partnership share.......................       1,116,107       2,052,025
Trust interest..........................         x99.99%         x99.99%
                                          --------------  --------------
Trust share.............................  $    1,115,995  $    2,051,820
                                          ==============  ==============

------------

(1) The amounts for the three months ended June 30, 1998 and 1997 represent
    actual production for the periods February 1997 through April 1998 and
    February 1996 through April 1997, respectively.

(2) Undistributed net loss represents negative Net Proceeds generated during the
    respective period. An undistributed net loss is carried forward and offset,
    in future periods, by positive Net Proceeds earned by the related Working
    Interest Owner(s). Undistributed net income represents positive Net
    Proceeds, generated during the respective period, that were applied to an
    existing loss carryforward. As of June 30, 1998, the loss carryforward was
    $1,545,348 ($386,337 net to the Trust).

                                       13
<PAGE>

                                                ROYALTY PROPERTIES
                                           SIX MONTHS ENDED JUNE 30,(1)
                                          ------------------------------
                                               1998            1997
                                          --------------  --------------
Crude oil and condensate (bbls).........         662,785         740,422
Natural gas and gas products (Mcf)......       2,825,074       3,648,437
Crude oil and condensate average price,
  per bbl...............................  $        15.66  $        21.19
Natural gas average price, per Mcf
  (excluding gas products)..............  $         2.52  $         2.91
Crude oil and condensate revenues.......  $   10,380,889  $   15,692,677
Natural gas and gas products revenues...       6,992,612      10,560,234
Production expenses.....................      (1,951,943)     (4,012,986)
Capital expenditures....................      (8,373,033)     (2,844,526)
Undistributed Net Loss (Income)(2)......        (136,438)     (3,571,111)
(Provision for) Refund of escrowed
  special costs.........................       5,290,681      (3,111,716)
                                          --------------  --------------
NET PROCEEDS............................      12,202,768      12,712,572
Royalty interest........................            x25%            x25%
                                          --------------  --------------
Partnership share.......................       3,050,692       3,178,143
Trust interest..........................         x99.99%         x99.99%
                                          --------------  --------------
Trust share.............................  $    3,050,387  $    3,177,825
                                          ==============  ==============

------------

(1) The amounts for the six months ended June 30, 1998 and 1997 represent actual
    production for the periods November 1997 through April 1998 and November
    1996 through April 1997 respectively.

(2) Undistributed net loss represents negative Net Proceeds generated during the
    respective period. An undistributed net loss is carried forward and offset,
    in future periods, by positive Net Proceeds earned by the related Working
    Interest Owner(s). Undistributed net income represents positive Net
    Proceeds, generated during the respective period, that were applied to an
    existing loss carryforward. As of June 30, 1998, the loss carryforward was
    $1,545,348 ($386,337 net to the Trust).

                                       14

<PAGE>
                          PART II -- OTHER INFORMATION

ITEM 6. EXHIBITS AND REPORTS ON FORM 8-K.

(A)  EXHIBITS

     (Asterisk indicates exhibit previously filed with the Securities and
Exchange Commission and incorporated herein by reference.)

<TABLE>
<CAPTION>
                                                                                                SEC FILE OR
                                                                                               REGISTRATION      EXHIBIT
                                                                                                  NUMBER         NUMBER
                                                                                               -------------     -------
<S>           <C>                                                                                 <C>               <C> 
              4(a)*     --  Trust Agreement dated as of January 1, 1983, among Tenneco
                            Offshore Company, Inc., Texas Commerce Bank National
                            Association, as corporate trustee, and Horace C. Bailey, Joseph
                            C. Broadus and F. Arnold Daum, as individual trustees (Exhibit
                            4(a) to Form 10-K for the year ended December 31, 1992 of TEL
                            Offshore Trust)................................................        0-6910           4(a)
              4(b)*     --  Agreement of General Partnership of TEL Offshore Trust
                            Partnership between Tenneco Oil Company and the TEL Offshore
                            Trust, dated January 1, 1983 (Exhibit 4(b) to Form 10-K for
                            year ended December 31, 1992 of TEL Offshore Trust)............        0-6910           4(b)
              4(c)*     --  Conveyance of Overriding Royalty Interests from Exploration I
                            to the Partnership (Exhibit 4(c) to Form 10-K for year ended
                            December 31, 1992 of TEL Offshore Trust).......................        0-6910           4(c)
              4(d)*     --  Amendments to TEL Offshore Trust Trust Agreement, dated
                            December 7, 1984 (Exhibit 4(d) to Form 10-K for year ended
                            December 31, 1992 of TEL Offshore Trust).......................        0-6910           4(d)
              4(e)*     --  Amendment to the Agreement of General Partnership of TEL
                            Offshore Trust Partnership, effective as of January 1, 1983
                            (Exhibit 4(e) to Form 10-K for year ended December 31, 1992 of
                            TEL Offshore Trust)............................................        0-6910           4(e)
              10(a)*    --  Purchase Agreement, dated as of December 7, 1984 by and between
                            Tenneco Oil Company and Tenneco Offshore II Company (Exhibit
                            10(a) to Form 10-K for year ended December 31, 1992, of TEL
                            Offshore Trust)................................................        0-6910          10(a)
              10(b)*    --  Consent Agreement, dated November 16, 1988, between TEL
                            Offshore Trust and Tenneco Oil Company (Exhibit 10(b) to Form
                            10-K for year ended December 31, 1988 of TEL Offshore Trust)...        0-6910          10(b)
              10(c)*    --  Assignment and Assumption Agreement, dated November 17, 1988,
                            between Tenneco Oil Company and TOC-Gulf of Mexico Inc.
                            (Exhibit 10(c) to Form 10-K for year ended December 31, 1988 of
                            TEL Offshore Trust)............................................        0-6910          10(c)
              10(d)*    --  Gas Purchase and Sales Agreement Effective September 1, 1993
                            between Tennessee Gas Pipeline Company and Chevron U.S.A.
                            Production Company (Exhibit 10(d) to Form 10-K for year ended
                            December 31, 1993 of TEL Offshore Trust).......................        0-6910          10(d)
              27(a)     --  Financial Data Schedule

</TABLE>
(B)  REPORTS ON FORM 8-K

     No reports on Form 8-K were filed with the Securities and Exchange
Commission during the second quarter of 1998.

                                       15
<PAGE>
                                   SIGNATURES

     Pursuant to the requirements of the Securities Exchange Act of 1934, the
Registrant has duly caused this report to be signed on its behalf by the
undersigned, thereunto duly authorized.

                                          TEL OFFSHORE TRUST

                                          By:  Chase Bank of Texas, National
                                               Association, Corporate Trustee


                                          By: /s/ PETE FOSTER
                                                  PETE FOSTER
                                                  SENIOR VICE PRESIDENT
                                                   AND TRUST OFFICER

Date: August 13, 1998

The Registrant, TEL Offshore Trust, has no principal executive officer,
principal financial officer, board of directors or persons performing similar
functions. Accordingly, no additional signatures are available and none have
been provided.

                                       16

<PAGE>
                                      TEL
                                    OFFSHORE
                                     TRUST
                               FEDERAL INCOME TAX
                                  INFORMATION
                                      1998
<PAGE>
                               TEL OFFSHORE TRUST
                                 EIN 76-6004064
                                   SCHEDULE B
                              SECOND QUARTER 1998
                            TAX INFORMATION PER UNIT
                               (4,751,510 UNITS)

<TABLE>
<CAPTION>
                                                   PARTNERSHIP
                                                 ITEMS PER UNIT                      TRUST ITEMS PER UNIT
                                          -----------------------------   -------------------------------------------
                                                          DEPLETION                                      **(INCREASE)
                                           ROYALTY       AS A PERCENT      INTEREST    ADMINISTRATIVE      DECREASE
              RECORD DATE                   INCOME     OF ROYALTY BASIS     INCOME        EXPENSE         IN RESERVE
----------------------------------------  ----------   ----------------   ----------   --------------    ------------
<S>   <C> <C>                               <C>          <C>                <C>           <C>                <C>     
March 31, 1998..........................    0.407111     7.9661%            0.003936      0.020072           0.022446
June 30, 1998...........................    0.234872     4.5958%            0.003679      0.024796          (0.000076)
September 30, 1998......................    0.000000     0.0000%            0.000000      0.000000          (0.000000)
December 31, 1998.......................    0.000000     0.0000%            0.000000      0.000000           0.000000
                                          ----------   ----------------   ----------   --------------    ------------
Year to date............................    0.641983    12.5619%            0.007615      0.044868           0.022370
                                          ==========   ================   ==========   ==============    ============
</TABLE>

                           SUMMARY OF TAXABLE INCOME

                                           PER UNIT
                                           ---------
Royalty Income..........................    0.641983
Interest Income.........................    0.007615
Depletion Deduction.....................   (0.020877)
Administrative Expense Deduction........   (0.044868)
(Increase)/Decrease in Reserve..........    0.022370
                                           ---------
Net Amount..............................    0.606223
                                           =========
<PAGE>
                        TAX BASIS OF UNITS AND ROYALTY*

Basis Assigned to TEL Offshore Trust
  Units -- 1/1/83.......................  $    6.750000
Basis Allocated to Offshore II Company
  (Sold December 17, 1984)..............      (0.120000)
                                          -------------
Royalty Basis 1/1/83....................       6.630000
Depletion Year 1983.....................      (0.769366)
                                          -------------
Royalty Basis 1/1/84....................       5.860634
Depletion Year 1984.....................      (1.203489)
                                          -------------
Royalty Basis 1/1/85....................       4.657145
Depletion Year 1985.....................      (1.126563)
                                          -------------
Royalty Basis 1/1/86....................       3.530582
Depletion Year 1986.....................      (0.555675)
                                          -------------
Royalty Basis 1/1/87....................       2.974907
Depletion Year 1987.....................      (1.424231)
                                          -------------
Royalty Basis 1/1/88....................       1.550676
Depletion Year 1988.....................      (0.384321)
                                          -------------
Royalty Basis 1/1/89....................       1.166355
Depletion Year 1989.....................      (0.241515)
                                          -------------
Royalty Basis 1/1/90....................       0.924840
Depletion Year 1990.....................      (0.242097)
                                          -------------
Royalty Basis 1/1/91....................       0.682743
Depletion Year 1991.....................      (0.092228)
                                          -------------
Royalty Basis 1/1/92....................       0.590515
Depletion Year 1992.....................      (0.058181)
                                          -------------
Royalty Basis 1/1/93....................       0.532334
Depletion Year 1993.....................      (0.079729)
                                          -------------
Royalty Basis 1/1/94....................       0.452605
Depletion Year 1994.....................      (0.148288)
                                          -------------
Royalty Basis 1/1/95....................       0.304317
Depletion Year 1995.....................      (0.064972)
                                          -------------
Royalty Basis 1/1/96....................       0.239345
Depletion Year 1996.....................      (0.024448)
                                          -------------
Royalty Basis 1/1/97....................       0.214897
Depletion Year 1997.....................      (0.048705)
                                          -------------
Royalty Basis 1/1/98....................       0.166192
Depletion Through Second Quarter 1998...      (0.020877)
                                          -------------
Royalty Basis 6/30/98...................  $    0.145315
                                          =============

------------

 * For Unit holders who acquired their Units in the initial distribution in
   January of 1983.

   For Unit holders acquiring Units other than in the initial distribution from
   Tenneco Offshore Company and prior to December 17, 1984, their royalty basis
   should be equal to 98.2533% of the purchase price of such Units, less
   depletion taken from the date of purchase.

   Unit holders who acquired their Units after December 17, 1984 will have a
   basis in the royalty equal to the purchase price of such Units, less
   depletion taken from the date of purchase.

** Increase or decrease in the reserve amount has no tax effect and is shown for
   information purposes only.